The signal detection theory evolved from the development of communications and radar equipment the first half of this century. It migrated to psychology, initially as part of sensation and perception, in the 50’s and 60’s as an attempt to understand some of the features of human behavior when detecting very faint stimuli that were not being explained by traditional theories of thresholds.
In my masters I have had participants look at pictures on a computer screen, in which half of the pictures contain face-like elements (stimuli) and the other half are pictures without face-like elements (control). The participant must make a decision, is there a face there or not. What makes this situation confusing and difficult is the presences of other mess that is similar to the stimuli (face-like elements). This mess is called noise.
What makes this different from traditional threshold theories is that the subject makes a decision, a cognitive act, as to whether the stimuli is present or not. This basic sensory act of determining if a stimulus occurred now is understood to have a cognitive component.
Now what can happen in this situation. If the face-like elements is present the person can decide that it is present or absent. These outcomes are called hits and misses. If the signal is absent the person can still decide that the signal is either present or absent. These are called false alarms or correct rejections (CR) respectively.
Read more on the subject here: http://www-psych.stanford.edu/~lera/psych115s/notes/signal/
Link to an excel-sheet with formulas here: d’_calculator